What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?

I believe this problem has been mentioned a few times in the literature, and has been solved for certain restrictions on the curve. For example if the curve has no four coplanar points then the maximal volume is achieved by one turn of a circular helix of height $\frac{1}{\sqrt{3}}$ and base radius $\frac{1}{\pi\sqrt{6}}$, this is due to Egervary. For more references see section A28 of "Unsolved problems in geometry" by H.T. Croft, K.J. Falconer, R.K. Guy. Melzak and Schoenberg have treated the corresponding problem for closed loops (Schoenberg has treated even dimensions), and have given answers under similar restrictions. In no case is a complete answer known. 


Here is an image of the optimal open convex curve. Taken from Open Problems from CCCG 2012, based on this paper, which cites Nudel'man (1975):



In the case of a closed curve my first wild (but educated :) guess would be: connect the following $\ 8\ $points (vertices of a cube but also belonging to a sphere) in the given cyclic order (of a maximal shift register): $$(a\ \!a\ \!a)\quad(a\ \!a\ +\!a)\quad(a\ +\!a\ +\!a)\quad(a\ +\!a\ \!a)$$ $$(+a\ +\!a\ \!a)\quad(+a\ +\!a\ +a)\quad(+a\ \!a\ +\!a)\quad(+a\ \!a\ \!a) $$ where $\ a\ $ is such that the length of the large arc which connects the consecutive points on the sphere which contains these $\ 8\ $ points is $\ \frac18$. EDIT Shooting from the hip cannot be that precise. Thus two related questions may help some:


